Short answers, pulled from the story.
The consensus problem requires agreement among a number of processes or agents on a single data value. This fundamental issue in distributed computing ensures overall system reliability even when some processes fail or behave unreliably.
Fischer, Lynch and Paterson published the famous 1985 FLP impossibility result which proves that deterministic algorithms for achieving consensus are impossible in asynchronous systems with at least one crash failure. This discovery derives from worst-case scheduling scenarios unlikely to occur except during adversarial situations like denial-of-service attacks.
Bitcoin introduced the first permissionless consensus protocol using proof of work where participants compete to solve cryptographic hash puzzles. Miners earn the right to commit blocks and receive rewards proportional to their invested computational effort through this probabilistic mechanism.
Proof of stake systems consume significantly less energy than proof of work because nodes compete to append blocks based on existing cryptocurrency allocated and locked for a time period. For example, bitcoin mining in 2018 consumed non-renewable energy sources similar to entire nations while Ethereum used power comparable to just under 205 average US households.
Herlihy's hierarchy defines the consensus number as the maximum number of processes that can reach consensus by a given object in a wait-free implementation. Read/write registers cannot solve consensus even in two-process systems while universal objects have an infinity consensus number capable of solving consensus among any number of processes.